102 research outputs found
Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory
In the present paper, we consider large scale nonsymmetric differential
matrix Riccati equations with low rank right hand sides. These matrix equations
appear in many applications such as control theory, transport theory, applied
probability and others. We show how to apply Krylov-type methods such as the
extended block Arnoldi algorithm to get low rank approximate solutions. The
initial problem is projected onto small subspaces to get low dimensional
nonsymmetric differential equations that are solved using the exponential
approximation or via other integration schemes such as Backward Differentiation
Formula (BDF) or Rosenbrok method. We also show how these technique could be
easily used to solve some problems from the well known transport equation. Some
numerical experiments are given to illustrate the application of the proposed
methods to large-scale problem
Tensorized block rational Krylov methods for tensor Sylvester equations
We introduce the definition of tensorized block rational Krylov subspaces and
its relation with multivariate rational functions, extending the formulation of
tensorized Krylov subspaces introduced in [Kressner D., Tobler C., Krylov
subspace methods for linear systems with tensor product structure, SIMAX,
2010]. Moreover, we develop methods for the solution of tensor Sylvester
equations with low multilinear or Tensor Train rank, based on projection onto a
tensor block rational Krylov subspace. We provide a convergence analysis, some
strategies for pole selection, and techniques to efficiently compute the
residual.Comment: 22 pages, 6 figures, 3 table
Krylov methods for large-scale modern problems in numerical linear algebra
Large-scale problems have attracted much attention in the last decades since
they arise from different applications in several fields. Moreover, the matrices that
are involved in those problems are often sparse, this is, the majority of their entries
are zero. Around 40 years ago, the most common problems related to large-scale and
sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors,
solving least square problems or computing singular value decompositions.
However, in the last years, large-scale and sparse problems of different natures have
appeared, motivating and challenging numerical linear algebra to develop effective
and efficient algorithms to solve them.
Common difficulties that appear during the development of algorithms for solving
modern large-scale problems are related to computational costs, storage issues and
CPU time, given the large size of the matrices, which indicate that direct methods
can not be used. This suggests that projection methods based on Krylov subspaces
are a good option to develop procedures for solving large-scale and sparse modern
problems.
In this PhD Thesis we develop novel and original algorithms for solving two
large-scale modern problems in numerical linear algebra: first, we introduce the
R-CORK method for solving rational eigenvalue problems and, second, we present
projection methods to compute the solution of T-Sylvester matrix equations, both
based on Krylov subspaces.
The R-CORK method is an extension of the compact rational Krylov method
(CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can
be expressed and linearized in certain particular ways and which include arbitrary
polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems.
The R-CORK method exploits the structure of the linearized problem by representing
the Krylov vectors in a compact form in order to reduce the cost of storage,
resulting in a method with two levels of orthogonalization. The first level of orthogonalization
works with vectors of the same size as the original problem, and the
second level works with vectors of size much smaller than the original problem. Since
vectors of the size of the linearization are never stored or orthogonalized, R-CORK
is more efficient from the point of view of memory and orthogonalization costs than
the classical rational Krylov method applied to the linearization. Moreover, since
the R-CORK method is based on a classical rational Krylov method, the implementation
of implicit restarting is possible and we present an efficient way to do it, that
preserves the compact representation of the Krylov vectors.
We also introduce in this dissertation projection methods for solving the TSylvester
equation, which has recently attracted considerable attention as a consequence
of its close relation to palindromic eigenvalue problems and other applications.
The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms
to solve these matrix equations for small- to medium- sized matrices. However,
developing numerical algorithms for solving large-scale T-Sylvester equations was a
completely open problem. In this thesis, we introduce several projection methods
based on block Krylov subspaces and extended block Krylov subspaces for solving
the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer
an intuition on the expected convergence of the algorithm based on block Krylov
subspaces and a clear guidance on which algorithm is the most convenient to use in
each situation.
All the algorithms presented in this thesis have been extensively tested, and the
reported numerical results show that they perform satisfactorily in practice.Adicionalmente se recibió ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix
Polynomials, Special Matrices, and Conditioning” (Ministerio de Economía y Competitividad de España, Número
de proyecto: MTM2012-32542) y “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de Economía y Competitividad de España,
Número de proyecto: MTM2015-65798-P), donde el investigador principal de ambos proyectos fue Froilán Martínez
Dopico.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: José Mas Marí.- Secretario: Fernando de Terán Vergara.- Vocal: José Enrique Román Molt
Rational Krylov for Stieltjes matrix functions: convergence and pole selection
Evaluating the action of a matrix function on a vector, that is , is an ubiquitous task in applications. When is large, one
usually relies on Krylov projection methods. In this paper, we provide
effective choices for the poles of the rational Krylov method for approximating
when is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is
equivalent, completely monotonic) and is a positive definite
matrix. Relying on the same tools used to analyze the generic situation, we
then focus on the case , and
obtained vectorizing a low-rank matrix; this finds application, for instance,
in solving fractional diffusion equation on two-dimensional tensor grids. We
see how to leverage tensorized Krylov subspaces to exploit the Kronecker
structure and we introduce an error analysis for the numerical approximation of
. Pole selection strategies with explicit convergence bounds are given also
in this case
Krylov subspace techniques for model reduction and the solution of linear matrix equations
This thesis focuses on the model reduction of linear systems and the solution of large
scale linear matrix equations using computationally efficient Krylov subspace techniques.
Most approaches for model reduction involve the computation and factorization of large
matrices. However Krylov subspace techniques have the advantage that they involve only
matrix-vector multiplications in the large dimension, which makes them a better choice
for model reduction of large scale systems. The standard Arnoldi/Lanczos algorithms are
well-used Krylov techniques that compute orthogonal bases to Krylov subspaces and, by
using a projection process on to the Krylov subspace, produce a reduced order model that
interpolates the actual system and its derivatives at infinity. An extension is the rational
Arnoldi/Lanczos algorithm which computes orthogonal bases to the union of Krylov
subspaces and results in a reduced order model that interpolates the actual system and
its derivatives at a predefined set of interpolation points. This thesis concentrates on the
rational Krylov method for model reduction.
In the rational Krylov method an important issue is the selection of interpolation points
for which various techniques are available in the literature with different selection criteria.
One of these techniques selects the interpolation points such that the approximation
satisfies the necessary conditions for H2 optimal approximation. However it is possible
to have more than one approximation for which the necessary optimality conditions are
satisfied. In this thesis, some conditions on the interpolation points are derived, that
enable us to compute all approximations that satisfy the necessary optimality conditions
and hence identify the global minimizer to the H2 optimal model reduction problem.
It is shown that for an H2 optimal approximation that interpolates at m interpolation
points, the interpolation points are the simultaneous solution of m multivariate polynomial
equations in m unknowns. This condition reduces to the computation of zeros of a
linear system, for a first order approximation. In case of second order approximation the
condition is to compute the simultaneous solution of two bivariate polynomial equations.
These two cases are analyzed in detail and it is shown that a global minimizer to the
H2 optimal model reduction problem can be identified. Furthermore, a computationally
efficient iterative algorithm is also proposed for the H2 optimal model reduction problem
that converges to a local minimizer.
In addition to the effect of interpolation points on the accuracy of the rational interpolating
approximation, an ordinary choice of interpolation points may result in a reduced
order model that loses the useful properties such as stability, passivity, minimum-phase and bounded real character as well as structure of the actual system. Recently in the
literature it is shown that the rational interpolating approximations can be parameterized
in terms of a free low dimensional parameter in order to preserve the stability of the
actual system in the reduced order approximation. This idea is extended in this thesis
to preserve other properties and combinations of them. Also the concept of parameterization
is applied to the minimal residual method, two-sided rational Arnoldi method
and H2 optimal approximation in order to improve the accuracy of the interpolating
approximation.
The rational Krylov method has also been used in the literature to compute low rank
approximate solutions of the Sylvester and Lyapunov equations, which are useful for
model reduction. The approach involves the computation of two set of basis vectors in
which each vector is orthogonalized with all previous vectors. This orthogonalization
becomes computationally expensive and requires high storage capacity as the number of
basis vectors increases. In this thesis, a restart scheme is proposed which restarts without
requiring that the new vectors are orthogonal to the previous vectors. Instead, a set of
two new orthogonal basis vectors are computed. This reduces the computational burden
of orthogonalization and the requirement of storage capacity. It is shown that in case
of Lyapunov equations, the approximate solution obtained through the restart scheme
approaches monotonically to the actual solution
Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations
We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples
On an integrated Krylov-ADI solver for large-scale Lyapunov equations
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to
solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efcient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems
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